CR transversality of holomorphic mappings between generic submanifolds in complex spaces

We show that a holomorphic mapping sending one generic submanifold into another of the same dimension is CR transversal to the target submanifold provided that the source manifold is of finite type and the map is of generic full rank. This result and its corollaries completely resolve two questions posed by Linda P. Rothschild and the first author in a paper from 2006.


Introduction
In this paper, we study CR transversality of holomorphic mappings sending a generic submanifold in C N into another of the same dimension. Recall that if U is an open subset of C N , H a holomorphic mapping U → C N , and M ′ a generic submanifold through a point p ′ := H(p) for some p ∈ U, then H is said to be CR transversal to M ′ at p if (1) T 1,0 p ′ M ′ + dH(T 1,0 p C N ) = T 1,0 p ′ C N , where T 0,1 M ′ = CT M ′ ∩ T 0,1 C N denotes the CR bundle on M ′ and T 1,0 M ′ = T 0,1 M ′ its complex conjugate. We remark that CR transversality of a holomorphic mapping is a property that in general is strictly stronger than that of transversality as a smooth mapping, i.e. CR transversality implies transversality but the converse does not hold in general. Transversality is an important and basic notion in geometry and analysis. For instance, an example of a transversality result in analysis is the classical Hopf Lemma for subharmonic functions in a smoothly bounded domain in R n (see e.g. [GT83]). The reader is referred to e.g. the paper [ER06] for further elaboration on the significance of tranversality in geometry and analysis. Also, the reader is referred to Section 2 for definitions and further explanation of the notions that appear in this introductory section.
The problem of CR transversality has been considered in various situations by many authors, see e.g. [Forn76], [Forn78], [P77], [BB82], [BR90], [BR93], [CR94], [CR98], [BHR95], [HP96], [ER06], [BER07] and the references therein. In this paper, we will consider the situation in which there is a generic submanifold M ⊂ U, of the same dimension as M ′ , such that p ∈ M and H(M) ⊂ M ′ . In [ER06], this same situation is considered and it is proved that the map H is CR transversal to M ′ at p ∈ M provided that M ′ is of The first author was partly supporting by the NSF grant DMS-0701121. The second author acknowledges a scholarship from the Vietnam Education Foundation for his graduate study at UC San Diego and a grant from NAFOSTED (Vietnam). finite type (in the sense of Kohn and Bloom-Graham) at p ′ := H(p) and the restriction of H to the Segre variety of M at p is finite (as a mapping into the Segre variety of M ′ at p ′ ). The condition on H is satisfied if, for instance, H itself is a finite mapping (C N , p) → (C N , p ′ ) (i.e. the inverse image H −1 (p ′ ) equals the singleton {p} as a germ of a variety at p). By combining several results including the one mentioned above, it is noted in [ER06] that if M and M ′ are hypersurfaces (i.e. submanifolds of real codimension one) and M is essentially finite at p, a stronger condition than finite type in the hypersurface case, then H is CR transversal to M ′ at p ′ under the assumption that Jac H ≡ 0, where Jac H denotes the Jacobian determinant of H and ≡ means "not identically zero"; we remark that Jac H ≡ 0 is strictly weaker than being a finite map. It was conjectured in [ER06] that the condition of essential finiteness of M at p could be weakened to finite type at p while maintaining the weak requirement Jac H ≡ 0 for the mapping. It was also conjectured that the condition Jac H ≡ 0 would be sufficient to conclude CR tranversality in the case of higher codimensional generic submanifolds M and M ′ provided that M is assumed to be of finite type and essentially finite at p. (For higher codimensional generic submanifolds, the conditions of being of finite type and essentially finite are unrelated, and the authors of [ER06] imposed both conditions on M in this conjecture to be on the safe side.) In this paper, we prove both of these conjectures. Indeed, we even prove that the condition of essential finiteness in the higher codimensional case is superfluous. More precisely, we have the following theorem.
Theorem 1.1. Let M, M ′ ⊂ C N be smooth generic submanifolds of the same dimension through p and p ′ respectively, and H : (C N , p) → (C N , p ′ ) a germ of a holomorphic mapping such that H(M) ⊂ M ′ . Assume that M is of finite type at p and Jac H ≡ 0. Then H is CR transversal to M ′ at p ′ .
We note that if M ′ is of finite type at p ′ and the restriction of H to the Segre variety of M at p is finite, then M is of finite type at p and Jac H ≡ 0 (see Proposition 2.3 in [ER06]). Thus, Theorem 1.1 also implies the result in [ER06] described above (Theorem 1.4 in [ER06]).
The condition that M is of finite type at p in Theorem 1.1 is optimal, as the following example (see Example 6.2 [ER06]) shows.
It is also clear that some a priori condition on the mapping H is necessary for the conclusion of Theorem 1.1 to hold, as is easily verified by simple examples. The reader is referred to e.g. Example 2.4 in [BER07] (in which one may consider M ⊂ C 2 as a finite type hypersurface in C 3 via a standard embedding) for an example of a mapping H (with Jac H ≡ 0) that is not CR transversal to M ′ at 0. If the source manifold M is holomorphically nondegenerate, then in fact the condition Jac H ≡ 0 is necessary for CR transversality to hold (see Proposition 3.6 below). We state here the following result, which is a direct consequence of Theorem 1.1 and Proposition 3.6. If we impose stronger conditions on M, such as essential finiteness or finite nondegeneracy in addition to finite type, then we also obtain stronger conclusions as in [ER06]. Indeed, by combining our Theorem 1.1 with Theorems 6.1 and 6.6 in [ER06], we obtain the following result; the notions of of essential finiteness and finite nondegeneracy will not play a role in any of the proofs in this paper and, hence, the reader is referred to [ER06] or the book [BER99a] for their definitions.
Theorem 1.4. Let M, M ′ ⊂ C N be smooth generic submanifolds through p and p ′ respectively, and H : (C N , p) → (C N , p ′ ) a germ of a holomorphic mapping such that H(M) ⊂ M ′ and Jac H ≡ 0 . If M is of finite type and essentially finite at p, then H is a finite map (C N , p) → (C N , p ′ ). If, in addition, M is finitely nondegenerate at p, then H is a local biholomorphism near p.
We mention that the notion of CR transversality can be defined also for CR mappings f : M → M ′ in terms of a formal power series expansion for f (see e.g. [ER06]). The analog of Theorem 1.1 holds for CR mappings whose differentials have generic maximal rank.
In the next section, we will briefly recall some basic definitions and facts. In Section 3, we will present the proof of Theorem 1.1. For further comments, examples and related results regarding this problem, we refer the reader to e.g. the papers [BR90] and [ER06].

Preliminaries
In this section, we will recall some basic definitions and facts about real submanifolds in complex spaces. For more detail and proofs of facts stated, we refer the reader to the book [BER99a]. Recall that a real submanifold M of codimension d in C N ( ∼ = R 2N ) is said to be generic if, for every p ∈ M, the submanifold M is defined locally near p by a defining equation ρ(Z,Z) = 0, where ρ = (ρ 1 , . . . ρ d ) is a smooth R d -valued function satisfying the following condition In particular, a generic submanifold M of codimension d in C N is a CR manifold of CR dimension n = N − d. If M is real-analytic and p ∈ M, then there are normal (local) coordinates Z = (z, w), where z = (z 1 , . . . , z n ) and w = (w 1 , . . . w d ), vanishing at p such that M is defined by the complex equation The fact that the d complex equations in (3) define a submanifold of real codimension d is equivalent to the identity here and in what follows, we use the following notation: if u(x) is an analytic function or formal power series in some set of variables x, thenū(x) is the function or power series given byū(x) := u(x).
A submanifold M is said to be of finite type (in the sense of Kohn and Bloom-Graham) at p if the (complex) Lie algebra g M generated by all smooth CR (or (0,1)) vector fields on M and their conjugates satisfies g M (p) = CT p M. Our proof of Theorem 1.1 will rely partly on a characterization of finite type due to Baouendi, Rothschild and the first author given in [BER96] (see also [BER03]) in term of the generic rank of the iterated Segre mappings; precise details will be given in Section 3. A generic, real-analytic submanifold M is said to be holomorphically nondegenerate at p if there is no (nontrivial) germ at p of a holomorphic vector field tangent to M near p. By a germ at p of a holomorphic vector field shall mean a vector field of the form where the ϕ j are germs at p of holomorphic functions. We shall consider the variablesz,w in (3) as an independent set of complex variables ξ = (χ, τ ) and thus the complexified equation We shall refer to M as the complexification of M. If M, M ′ ⊂ C N are generic, real analytic submanifolds of codimension d, given in normal coordinates by w = Q(z,z,w) and w ′ = Q ′ (z ′ ,z ′ ,w ′ ), and H(z, w) = (F (z, w), G(z, w)) is a holomorphic mapping (C N , 0) → (C N , 0) sending M into M ′ , then there is a d × d matrix a(z, w, χ, τ ) of holomorphic functions in a neighborhood of the origin in C N Z × C N ξ such that the following identity holds: Equivalently, the complexified mapping H(z, w, χ, τ ) := (H(z, w),H(χ, τ )) sends M into M ′ (with the obvious notation). It is straightforward to verify that H is CR transversal to M ′ at 0 if and only if det a(0) = 0 (cf. [BER07]). It is convenient to relax the convergence properties of the map and defining equations.  C N , 0), sending M into M ′ ; the reader is e.g. referred to [BER99b] for this (fairly obvious) construction. It is also straightforward to verify that the holomorphic mapping H sending the smooth manifold M into the smooth manifold M ′ is CR transversal to M ′ at p if and only if the formal manifolds and mapping satisfy (6) with det a(0) = 0. Thus, to prove Theorem 1.1 it suffices to consider formal manifolds M and M ′ through 0 in C N , a formal holomorphic mapping H : (C N , 0) → (C N , 0) sending M into M ′ , i.e. satisfying (6), and prove that the matrix a(z, w, χ, τ ) in (6) satisfies det a(0) = 0; the reader is also referred to [ER06] for more details on this reduction to the formal case. In what follows, we shall consider only formal mappings and formal generic submanifolds.

Proof of the main theorem and corollaries
In this section, M and M ′ are generic formal submanifolds through 0, given in normal coordinates by w = Q(z,z,w) and w ′ = Q ′ (z ′ ,z ′ ,w ′ ), and H(z, w) = (F (z, w), G(z, w)) a formal mapping sending M into M ′ . We may consider H andH as formal maps from C N Z × C N ξ in the obvious way: H(z, w, χ, τ ) = H(z, w) H(z, w, χ, τ ) =H(χ, τ ) We begin with the following lemma.
Lemma 3.1. Let M and M ′ be generic formal submanifolds given in normal coordinates by w = Q(z,z,w) and w ′ = Q ′ (z ′ ,z ′ ,w ′ ) and H = (F, G) : (C N , 0) → (C N , 0) a formal mapping sending M into M ′ , that is where a(z, w, χ, τ ) is a d × d matrix of formal power series in C[[z, w, χ, τ ]]. Then, there are d × d matrices C(z, χ, τ ) and E(z, χ, τ ) of formal power series in C[[z, χ, τ ]] such that the following holds where I d denotes the d × d identity matrix.
We now factor both sides of (26) into products of irreducible elements in the unique factorization domain C[[z, w, χ]]. Since the left hand side of (26) is a non-trivial formal power series in the ring C[[z, w]] ⊂ C[[z, w, χ]], its factorization involves factors that are power series in z and w only. Thus, by the uniqueness of the factorization, we obtain det a(z, w, χ,Q(χ, z, w)) = u(z, w, χ)b(z, w).
We will need the following lemma whose proof may be found, e.g., in [BER99a], Proposition 5.2.3.
We shall now prove our main result.
For general case, we will need the iterated Segre mappings introduced in [BER96] (see also [BER03], [ER06]). For a positive integer k, the kth Segre mapping of M at 0 is the mapping v k : C kn → C N defined by (32) C kn ∋ t = (t 1 , . . . , t k ) → v k (t) := (t k , u k (t 1 , . . . , t k )), where u k : C kn → C d is given inductively by The crucial property of the Segre mappings needed here is the result (see [BER96], [BER99a], [BER99b], [BER03]) that M is of finite type at 0 if and only if the maps v k has generic rank N for k large enough (k ≥ d + 1 suffices). Thus, by Lemma 3.4, the following lemma implies that det H Z ≡ 0, which is a contradiction and completes the proof of Theorem 1.1.
Lemma 3.5. For every j ≥ 0, the following holds.
Proof. We may consider b(z, w) and c(χ, τ ) in Lemmas 3.2 and 3.3 as power series in (z, w, χ, τ ) by b(z, w, χ, τ ) = b(z, w) and c(z, w, χ, τ ) = c(χ, τ ). Since the complexification M of M is parametrized by it follows from (27) that b ∼ = c on M, where we use the notation α ∼ = β to mean α = γβ for some unit γ. Now, another crucial property of the Segre mappings v k (see e.g. [BER99b]) is that (v k+1 , v k ) ∈ M and (v k−1 , v k ) ∈ M for every k. Consequently, equation (27) implies that By induction, we obtain, for every positive integer j, This completes the proof of Lemma 3.5.
We may now prove Theorem 1.4.
Proof of Theorem 1.4. Since Jac H ≡ 0 and M is of finite type at p, it follows from Proposition 2.3 in [ER06] that M ′ is of finite type at p ′ . Also, by Theorem 1.1, H is CR transversal to M ′ at p and thus, in particular, transversally regular to M ′ (see [ER06]). Consequently, by Theorem 6.1 in [ER06], if M is also essentially finite at p, then H is finite mapping and M ′ is essentially finite at p ′ . If, in addition, M is finitely nondegenerate at p, then Theorem 6.6 in [ER06] asserts that H is local biholomorphism near p. Now, to prove Theorem 1.3, we shall need the following proposition. Proof. The idea of the proof is taken from [A08]. Assume, in order to reach a contradiction, that Jac H ≡ 0. Then, there is a non trivial N-vector U(z, w) with components in the field of fractions of C  (F (χ, τ ), F (z, w), G(z, w)) =ā(χ, τ, z, w)(τ −Q(χ, z, w)).
Consequently, we must also have L ā(χ, τ, z, w)(τ −Q(χ, z, w)) ≡ 0. In other words, It follows that L is tangent to M. This is a contradiction since M is holomorphically nondegenerate. The proof is complete.
Theorem 1.3 is now a direct consequence of Theorem 1.1 and Proposition 3.6.